2020 Sample Specialist Mathematics Examination Paper BK2

SAMPLE2

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Specialist MathematicsNovember 2020 sample paper

Question booklet 2• Questions 8 to 10 (46 marks)• Answer all questions• Write your answers in this question booklet• You may write on page 12 if you need more space• Allow approximately 65 minutes• Approved calculators may be used — complete the box below

The purpose of this sample paper is to show the structure of the 130-minute Specialist Mathematics examination and the style of questions that might be used. The examination will consist of questions that assess a selection of the key questions and key concepts from across the six topics.

This sample Specialist Mathematics paper shows the format of the examination from November 2020.

SAMPLE

page 2 of 12

Question 8 (16 marks)

Consider the system of equations shown below.

x yx yx y

2 2 42 2 5

3 2 2 8

z

z

z

(a) (i) Write this system as an augmented matrix.

(1 mark)

(ii) Stating all row operations, show that this system of equations has the following solutions:

x t2 2 y t1 2z t

where t is a real parameter.

(3 marks)


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page 3 of 12 PLEASE TURN OVER

(b) Consider three planes in space, P1, P2 and P3, defined by the system of equations below.

P x yP x yP x y

1

2

3

2 2 42 2 5

3 2 2 8

:::

z

z

z

(i) Using the information given in part (a)(ii), show that the points A B2 1 0 0 3 1, , and , , are common to all three planes.

(2 marks)

(ii) Show that P1 and P2 are perpendicular.

(2 marks)


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page 4 of 12

(c) Figure 8 shows the point C 0 6 2, , on P3, the point D 12 4 0, , on P1, and the line l through C and D, intersecting P2 at the point E.

P2P3

P1

A

BE

D

C

l

Figure 8

(i) Find, in parametric form, the equation of l.

(2 marks)

(ii) Find the coordinates of E.

(2 marks)

(iii) Find the distance from E to P1.

(2 marks)


SAMPLE


(d) The equations of P1 and P3 are used to model two hillsides that meet at a river, as shown in Figure 9.

P x y1 2 2 4: z

P x y3 3 2 2 8: z

The river is modelled by the line where the two planes meet. A straight bridge, modelled by l, connects C 0 6 2, , to D 12 4 0, , .

D

P1 P3

CEl

Figure 9

The point E, on the bridge, must be at least 1 unit from P1 and at least 1 unit from P3.Does the model satisfy this condition? Show your calculations.

(2 marks)


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page 6 of 12


(a) (i) Solve the equation z3 1. Write the three solutions in polar form.

(2 marks)

(ii) Draw the solutions on the Argand diagram in Figure 10, labelling each solution in an anticlockwise direction 1, 2 , and 3, where 1 is real.

Re z

Im z

Figure 10 (2 marks)

(iii) Explain why 2 lies on the line with equation z z1 3 .

(1 mark)

(iv) Evaluate 1 2 3.

(1 mark)


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(v) Evaluate 1 2 3 .

(1 mark)

(vi) Find the exact value of 2 3 .

(1 mark)

(b) Consider the equation w 1 13 .

(i) Show that the solutions to this equation are w1 11 , w2 21 , and w3 31 , where 1, 2 , and 3 are the solutions to the equation z3 1, as in part (a).

(1 mark)

(ii) Draw the solutions of w 1 13 on the Argand diagram in Figure 11.

Re z

Im z

Figure 11 (2 marks)


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page 8 of 12

(iii) Show that w2 and w3 lie on the line Re z12

.

(1 mark)

(iv) Evaluate w w w1 2 3 .

(1 mark)

(v) Find the exact value of w w w w w w1 2 2 3 3 1 .

(2 marks)


SAMPLE



(a) In an experiment one type of bacterium, called alpha, was grown in a Petri dish.

The rate of change of the area in the Petri dish that was covered by alpha bacteria can be modelled by the differential equation

ddAt

A A12

5050

where A is area in cm2 and t is time in days.

(i) At t 0, the area in the Petri dish that was covered by alpha bacteria was 1 cm2.

On the slope field in Figure 12, draw the solution curve.

2 4 6 8 10 12 14 16 18 20

8

4

12

20

28

36

44

16

24

32

40

48

t

A

Figure 12 (3 marks)

(ii) Show that 5050

1 150A A A A

.

(1 mark)

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SAMPLE

page 10 of 12

(iii) Use integration to solve the differential equation

ddAt

A A12

5050

with initial condition A 0 1, and show that the area covered by alpha bacteria can be modelled by the equation below.

Ae t50

1 49 0 5.

(5 marks)

(iv) State the maximum area that is available for bacterial growth.

(1 mark)


SAMPLE


(b) When t 1, a different type of bacterium, called beta, was accidentally introduced to the Petri dish. Beta bacteria grow more quickly than alpha bacteria. The area, B, in the Petri dish that was covered by beta bacteria can be modelled by

Be d

501 79 5

where d is time in days after the beta bacterium was introduced.

Figure 13 shows the graph for the area in the Petri dish covered by beta bacteria d days after the beta bacterium was introduced.

10

20

1 3 4 5

30

40

50

B

d

Figure 13

(i) Find d and B when the rate of growth of beta bacteria was at its greatest.

(2 marks)

(ii) Find the area of the Petri dish covered by alpha bacteria when the rate of growth of beta bacteria is at its greatest.

(2 marks)

(iii) When the Petri dish is completely covered by bacteria, which type of bacterium is likely to cover more area of the Petri dish?

(1 mark)


SAMPLE

page 12 of 12 — end of booklet

You may write on this page if you need more space to finish your answers to any of the questions in Question booklet 2. Make sure to label each answer carefully (e.g. 10(b)(ii) continued).

2020 Sample Specialist Mathematics Examination Paper BK2

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